.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/beams/plot_04_winkler_influence_lines.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_structural_engineering_beams_plot_04_winkler_influence_lines.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_structural_engineering_beams_plot_04_winkler_influence_lines.py:


Winkler's influence lines (1867)
================================

A railway bridge must be checked for a train in every position. Emil
Winkler (1867-1868), and Otto Mohr independently (1868), plotted the value
of a response, such as the moment at one section, against the position of a
unit load: the *influence line*
(:func:`~engineeringkit.structural_engineering.influence_line`). The
response to any train is then the sum of the axle loads times the
ordinates under them, and the worst position can be read off the plot. The
example finds the worst position of a four-axle locomotive on a 20 m span
and checks it against a brute-force superposition of point-load solutions.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import SimplySupportedBeam, influence_line








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Influence lines for a section at the quarter span
-------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-36

.. code-block:: Python


    L, section = 20.0, 5.0
    xi = np.linspace(0, L, 401)
    fig, axes = plt.subplots(3, 1, sharex=True, figsize=(6, 6))
    for ax, quantity, unit in zip(axes, ["left_reaction", "shear", "moment"], ["[-]", "[-]", "[m]"]):
        ax.plot(xi, influence_line(L, xi, quantity, at=section))
        ax.axhline(0, color="gray", lw=0.5)
        ax.set_ylabel(f"{quantity} {unit}")
    axes[-1].set_xlabel("unit-load position [m]")
    fig.tight_layout()




.. image-sg:: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_04_winkler_influence_lines_001.png
   :alt: plot 04 winkler influence lines
   :srcset: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_04_winkler_influence_lines_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 37-39

Moving a locomotive across the span
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-55

.. code-block:: Python


    axles = np.array([0.0, 1.8, 3.6, 5.4])  # axle offsets from the front axle [m]
    weights = np.array([200e3, 200e3, 200e3, 200e3])  # [N]
    fronts = np.linspace(0, L + axles[-1], 1001)


    def section_moment(front):
        pos = front - axles
        on = (pos >= 0) & (pos <= L)
        return float(np.sum(weights[on] * influence_line(L, pos[on], "moment", at=section)))


    by_influence = np.array([section_moment(f) for f in fronts])
    best = fronts[np.argmax(by_influence)]
    print(f"worst front-axle position {best:.2f} m, moment at x = {section} m: {by_influence.max() / 1e3:.0f} kN m")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    worst front-axle position 9.80 m, moment at x = 5.0 m: 2460 kN m




.. GENERATED FROM PYTHON SOURCE LINES 56-57

Check against superposed point-load solutions of the beam itself.

.. GENERATED FROM PYTHON SOURCE LINES 57-73

.. code-block:: Python


    beam = SimplySupportedBeam(L, 200e9, 0.05)
    n = 801
    k = int(round(section / L * (n - 1)))
    direct = sum(beam.solve_point_load(Wt, position=best - a, n=n).moment[k] for a, Wt in zip(axles, weights) if 0 <= best - a <= L)
    print(f"by superposition: {direct / 1e3:.0f} kN m")
    assert np.isclose(direct, by_influence.max())

    fig, ax = plt.subplots(figsize=(6, 3.5))
    ax.plot(fronts, by_influence / 1e3)
    ax.axvline(best, color="C3", ls="--", label="worst position")
    ax.set(xlabel="front-axle position [m]", ylabel="moment at quarter span [kN m]", title="Four-axle locomotive")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_04_winkler_influence_lines_002.png
   :alt: Four-axle locomotive
   :srcset: /api/gallery/structural_engineering/beams/images/sphx_glr_plot_04_winkler_influence_lines_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    by superposition: 2460 kN m





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_beams_plot_04_winkler_influence_lines.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/structural_engineering/beams/plot_04_winkler_influence_lines.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_04_winkler_influence_lines.ipynb <plot_04_winkler_influence_lines.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_04_winkler_influence_lines.py <plot_04_winkler_influence_lines.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_winkler_influence_lines.zip <plot_04_winkler_influence_lines.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_